Editing CW complex (section)

==Definition==

=== CW complex ===
A '''CW complex''' is constructed by taking the union of a sequence of topological spaces <math display="block">\emptyset = X_{-1} \subset X_0 \subset X_1 \subset \cdots</math> such that each <math>X_k</math> is obtained from <math>X_{k-1}</math> by gluing copies of k-cells <math>(e^k_\alpha)_\alpha</math>, each homeomorphic to the open <math>k</math>-[[ball (mathematics)|ball]] <math>B^k</math>, to <math>X_{k-1}</math> by continuous gluing maps <math>g^k_\alpha: \partial e^k_\alpha \to X_{k-1}</math>. The maps are also called [[attaching map]]s. Thus as a set, <math>X_k = X_{k-1} \sqcup_{\alpha} e^k_\alpha</math>.

Each <math>X_k</math> is called the '''k-skeleton''' of the complex.

The topology of <math>X = \cup_{k} X_k</math> is '''weak topology''': a subset <math>U\subset X</math> is open [[iff]] <math>U\cap X_k</math> is open for each k-skeleton <math>X_k</math>.

In the language of category theory, the topology on <math>X</math> is the [[direct limit]] of the diagram <math display="block">X_{-1} \hookrightarrow X_0 \hookrightarrow X_1 \hookrightarrow \cdots</math>The name "CW" stands for "closure-finite weak topology", which is explained by the following theorem:

{{Math theorem
| name = Theorem
| note = 
| math_statement = A [[Hausdorff space]] ''X'' is homeomorphic to a CW complex iff there exists a [[partition of a set|partition]] of ''X'' into "open cells" <math>e^k_\alpha</math>, each with a corresponding closure (or "closed cell") <math>\bar{e}^k_\alpha := cl_X(e^k_\alpha)</math> that satisfies:

* For each <math>e^k_\alpha</math>, there exists a [[Continuous function#Continuous functions between topological spaces|continuous surjection]] <math>g_\alpha^k: D^k \to \bar{e}^k_\alpha</math> from the <math>k</math>-dimensional closed ball such that
** The restriction to the open ball <math>g_\alpha^k: B^k\to e^k_\alpha </math> is a [[homeomorphism]].
** (closure-finiteness) The image of the boundary <math>g^k_\alpha(\partial D^k) </math> is covered by a finite number of closed cells, each having cell dimension less than k.
* (weak topology) A subset of ''X'' is [[Closed set|closed]] if and only if it meets each closed cell in a closed set.
}}

This partition of ''X'' is also called a '''cellulation'''.

==== The construction, in words ====
The CW complex construction is a straightforward generalization of the following process:
* A 0-''dimensional CW complex'' is just a set of zero or more discrete points (with the [[Discrete space|discrete topology]]).
* A 1-''dimensional CW complex'' is constructed by taking the [[Disjoint union (topology)|disjoint union]] of a 0-dimensional CW complex with one or more copies of the [[unit interval]]. For each copy, there is a map that "[[Gluing (topology)|glues]]" its boundary (its two endpoints) to elements of the 0-dimensional complex (the points). The topology of the CW complex is the topology of the [[Quotient space (topology)|quotient space]] defined by these gluing maps.
* In general, an ''n-dimensional CW complex'' is constructed by taking the disjoint union of a ''k''-dimensional CW complex (for some <math>k<n</math>) with one or more copies of the [[Ball (mathematics)|''n''-dimensional ball]]. For each copy, there is a map that "glues" its boundary (the <math>(n-1)</math>-dimensional [[N-sphere|sphere]]) to elements of the <math>k</math>-dimensional complex. The topology of the CW complex is the [[quotient topology]] defined by these gluing maps.
* An ''infinite-dimensional CW complex'' can be constructed by repeating the above process countably many times. Since the topology of the union <math>\cup_k X_k</math> is indeterminate, one takes the direct limit topology, since the diagram is highly suggestive of a direct limit. This turns out to have great technical benefits.

===Regular CW complexes===
A '''regular CW complex''' is a CW complex whose gluing maps are homeomorphisms. Accordingly, the partition of ''X'' is also called a '''regular cellulation'''.

A [[Loop (graph theory)|loopless]] graph is represented by a regular 1-dimensional CW-complex. A [[graph embedding|closed 2-cell graph embedding]] on a [[surface]] is a regular 2-dimensional CW-complex. Finally, the 3-sphere regular cellulation conjecture claims that every [[k-vertex-connected graph|2-connected graph]] is the 1-skeleton of a regular CW-complex on the [[3-sphere|3-dimensional sphere]].<ref>{{cite conference|title=The 3-Sphere Regular Cellulation Conjecture|first=Sergio|last=De Agostino|conference=International Workshop on Combinatorial Algorithms|year=2016 |url=https://twiki.di.uniroma1.it/pub/Users/SergioDeAgostino/DeAgostino.pdf}}</ref>

===Relative CW complexes===
Roughly speaking, a ''relative CW complex'' differs from a CW complex in that we allow it to have one extra building block that does not necessarily possess a cellular structure. This extra-block can be treated as a (−1)-dimensional cell in the former definition.<ref>{{cite book |last1=Davis |first1=James F. |title=Lecture Notes in Algebraic Topology |last2=Kirk |first2=Paul |date=2001 |publisher=American Mathematical Society |location=Providence, R.I.}}</ref><ref>{{Cite web |title=CW complex in nLab |url=https://ncatlab.org/nlab/show/CW+complex}}</ref><ref>{{Cite web |title=CW-complex - Encyclopedia of Mathematics |url=https://www.encyclopediaofmath.org/index.php/CW-complex}}</ref>